Logarithmic Identities

In mathematics, many logarithmic identities exist. The following is a compilation of the notable of these, many of which are used for computational purposes.

Trivial identities

:

Explanations

By definition, we know that:
:$\color \log \color_b \color (\colory\color) = \colorx\color \iff \colorb\color \color^x\color = \colory\color$,
where $\colorb\color \neq 0$.

Setting $\colorx\color = 0$,
we can see that:
$\colorb\color \color^x\color = \colory\color \iff \colorb\color \color^\color = \colory\color \iff \color1\color = \colory\color \iff \colory\color = \color1\color$. So, substituting these values into the formula, we see that:
$\color \log \color_b \color (\colory\color) = \colorx\color \iff \color \log \color_b \color (\color1\color) = \color0\color$
, which gets us the first property.

Setting $\colorx\color = 1$,
we can see that:
$\colorb\color \color^x\color = \colory\color \iff \colorb\color \color^\color = \colory\color \iff \colorb\color = \colory\color \iff \colory\color = \colorb\color$. So, substituting these values into the formula, we see that:
$\color \log \color_b \color (\colory\color) = \colorx\color \iff \color \log \color_b \color (\colorb\color) = \color1\color$
, which gets us the second property.

Many mathematical identities are called trivial , only because they are relatively simple (typically from the perspective of an experienced mathematician).
This is not to say that calling an identity or formula trivial means that it's not important.

Cancelling exponentials

Logarithms and exponentials with the same base cancel each other. This is true because logarithms and exponentials are inverse operations—much like the same way multiplication and division are inverse operations, and addition and subtraction are inverse operations.

:$b^ = x\text\mbox_b(\log_b(x)) = x$

:$\log_b(b^x) = x\text\log_b(\mbox_b(x)) = x$

Both of the above are derived from the following two equations that define a logarithm:
(note that in this explanation, the variables of $\colorx\color$ and $x$ may not be referring to the same number)

:$\color \log \color_b \color (\colory\color) = \colorx\color \iff \colorb\color \color^x\color = \colory\color$

Looking at the equation $\colorb\color \color^x\color = \colory\color$, and substituting the value for $\colorx\color$ of
$\color \log \color_b \color (\colory\color) = \colorx\color$, we get the following equation:
$\colorb\color \color^x\color = \colory\color \iff \colorb\color \color^\color = \colory\color \iff \colorb\color \color^\color = \colory\color$
, which gets us the first equation.
Another more rough way to think about it is that $\colorb\color \color^\color = \colory\color$,
and that that "$\color$" is $\color \log \color_b \color (\colory\color)$.

Looking at the equation $\color \log \color_b \color (\colory\color) = \colorx\color$
, and substituting the value for $\colory\color$ of $\colorb\color \color^x\color = \colory\color$, we get the following equation:
$\color \log \color_b \color (\colory\color) = \colorx\color \iff \color \log \color_b \color (\color b^x\color) = \colorx\color \iff \color \log \color_b \color (\color) = \colorx\color$
, which gets us the second equation.
Another more rough way to think about it is that $\color \log \color_b \color (\color\text\color) = \colorx\color$,
and that that something "$\color\text$" is $\colorb\color \color^x\color$.

Using simpler operations

Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding. These are often known as logarithmic properties, which are documented in the table below. The first three operations below assume that and/or , so that and . Derivations also use the log definitions and .

:

Where $b$, $x$, and $y$ are positive real numbers and $b \ne 1$, and $c$ and $d$ are real numbers.

The laws result from canceling exponentials and the appropriate law of indices. Starting with the first law:

:$xy = b^ b^ = b^ \Rightarrow \log_b(xy) = \log_b(b^) = \log_b(x) + \log_b(y)$

The law for powers exploits another of the laws of indices:

:$x^y = (b^)^y = b^ \Rightarrow \log_b(x^y) = y \log_b(x)$

The law relating to quotients then follows:

:$\log_b \bigg(\frac\bigg) = \log_b(x y^) = \log_b(x) + \log_b(y^) = \log_b(x) - \log_b(y)$

:$\log_b \bigg(\frac\bigg) = \log_b(y^) = - \log_b(y)$

Similarly, the root law is derived by rewriting the root as a reciprocal power:

:\log_b(\sqrt ) = \log_b(x^) = \frac\log_b(x)

Derivations of product, quotient, and power rules

These are the three main logarithm laws/rules/principles, from which the other properties listed above can be proven. Each of these logarithm properties correspond to their respective exponent law, and their derivations/proofs will hinge on those facts. There are multiple ways to derive/prove each logarithm law – this is just one possible method.

Logarithm of a product

To state the logarithm of a product law formally:

:$\forall b \in \mathbb_+, b \neq 1, \forall x, y, \in \mathbb_+, \log_b(xy) = \log_b(x) + \log_b(y)$

Derivation:

Let $b \in \mathbb_+$, where $b \neq 1$,
and let $x, y \in \mathbb_+$. We want to relate the expressions $\log_b(x)$ and $\log_b(y)$. This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to $\log_b(x)$ and $\log_b(y)$ quite often, we will give them some variable names to make working with them easier: Let $m = \log_b(x)$, and let $n = \log_b(y)$.

Rewriting these as exponentials, we see that $m = \log_b(x) \iff b^m = x$ and $n = \log_b(y) \iff b^n = y$. From here, we can relate $b^m$ (i.e. $x$) and $b^n$ (i.e. $y$) using exponent laws as

:$xy = (b^m)(b^n) = b^m \cdot b^n = b^$

To recover the logarithms, we apply $\log_b$ to both sides of the equality.

:$\log_b(xy) = \log_b(b^)$

The right side may be simplified using one of the logarithm properties from before: we know that $\log_b(b^) = m + n$, giving

:$\log_b(xy) = m + n$

We now resubstitute the values for $m$ and $n$ into our equation, so our final expression is only in terms of $x$, $y$, and $b$.

:$\log_b(xy) = \log_b(x) + \log_b(y)$

This completes the derivation.

Logarithm of a quotient

To state the logarithm of a quotient law formally:

:$\forall b \in \mathbb_+, b \neq 1, \forall x, y, \in \mathbb_+, \log_b \left( \frac \right) = \log_b(x) - \log_b(y)$

Derivation:

Let $b \in \mathbb_+$, where $b \neq 1$,
and let $x, y \in \mathbb_+$.

We want to relate the expressions $\log_b(x)$ and $\log_b(y)$. This can be done more easily by rewriting in terms of exponentials, whose properties we already know. Additionally, since we are going to refer to $\log_b(x)$ and $\log_b(y)$ quite often, we will give them some variable names to make working with them easier: Let $m = \log_b(x)$, and let $n = \log_b(y)$.

Rewriting these as exponentials, we see that: $m = \log_b(x) \iff b^m = x$ and $n = \log_b(y) \iff b^n = y$. From here, we can relate $b^m$ (i.e. $x$) and $b^n$ (i.e. $y$) using exponent laws as

:$\frac = \frac = \frac = b^$

To recover the logarithms, we apply $\log_b$ to both sides of the equality.

:$\log_b \left( \frac \right) = \log_b \left( b^ \right)$

The right side may be simplified using one of the logarithm properties from before: we know that $\log_b(b^) = m - n$, giving

:$\log_b \left( \frac \right) = m -n$

We now resubstitute the values for $m$ and $n$ into our equation, so our final expression is only in terms of $x$, $y$, and $b$.

:$\log_b \left( \frac \right) = \log_b(x) - \log_b(y)$

This completes the derivation.

Logarithm of a power

To state the logarithm of a power law formally,

:$\forall b \in \mathbb_+, b \neq 1, \forall x \in \mathbb_+, \forall r \in \mathbb, \log_b(x^r) = r\log_b(x)$

Derivation:

Let $b \in \mathbb_+$, where $b \neq 1$, let $x\in \mathbb_+$, and let $r \in \mathbb$. For this derivation, we want to simplify the expression $\log_b(x^r)$. To do this, we begin with the simpler expression $\log_b(x)$. Since we will be using $\log_b(x)$ often, we will define it as a new variable: Let $m = \log_b(x)$.

To more easily manipulate the expression, we rewrite it as an exponential. By definition, $m = \log_b(x) \iff b^m = x$, so we have

:$b^m = x$

Similar to the derivations above, we take advantage of another exponent law. In order to have $x^r$ in our final expression, we raise both sides of the equality to the power of $r$:

:$\begin (b^m)^r &= (x)^r \\ b^ &= x^r \end$

where we used the exponent law $(b^m)^r = b^$.

To recover the logarithms, we apply $\log_b$ to both sides of the equality.

:$\log_b(b^) = \log_b(x^r)$

The left side of the equality can be simplified using a logarithm law, which states that $\log_b(b^) = mr$.

:$mr = \log_b(x^r)$

Substituting in the original value for $m$, rearranging, and simplifying gives

:$\begin \left( \log_b(x) \right)r &= \log_b(x^r) \\ r\log_b(x) &= \log_b(x^r) \\ \log_b(x^r) &= r\log_b(x) \end$

This completes the derivation.

Changing the base

To state the change of base logarithm formula formally:

:$\forall a, b \in \mathbb_+, a, b \neq 1 \forall x \in \mathbb_+, \log_b(x) = \frac$

This identity is useful to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log₁₀, but not all calculators have buttons for the logarithm of an arbitrary base.

Proof/derivation

Let $a, b \in \mathbb_+$, where $a, b \neq 1$ Let $x \in \mathbb_+$. Here, $a$ and $b$ are the two bases we will be using for the logarithms. They cannot be 1, because the logarithm function is not well defined for the base of 1. The number $x$ will be what the logarithm is evaluating, so it must be a positive number. Since we will be dealing with the term $\log_b(x)$ quite frequently, we define it as a new variable: Let $m = \log_b(x)$.

To more easily manipulate the expression, it can be rewritten as an exponential.

:$b^m = x$

Applying $\log_a$ to both sides of the equality,

:$\log_a(b^m) = \log_a(x)$

Now, using the logarithm of a power property, which states that $\log_a(b^m) = m\log_a(b)$,

:$m\log_a(b) = \log_a(x)$

Isolating $m$, we get the following:

:$m = \frac$

Resubstituting $m = \log_b(x)$ back into the equation,

:$\log_b(x) = \frac$

This completes the proof that $\log_b(x) = \frac$.

This formula has several consequences:

:$\log_b a = \frac 1$

:$\log_ a =$

:$b^ = d^$

:$-\log_b a = \log_b \left(\right) = \log_ a$

:$\log_a_1 \,\cdots\, \log_a_n = \log_a_1\, \cdots\, \log_a_n,$

where $\pi$ is any permutation of the subscripts 1, ..., ''n''. For example

:$\log_b w\cdot \log_a x\cdot \log_d c\cdot \log_d z = \log_d w\cdot \log_b x\cdot \log_a c\cdot \log_d z.$

Summation/subtraction

The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:

:

Note that the subtraction identity is not defined if $a=c$, since the logarithm of zero is not defined. Also note that, when programming, $a$ and $c$ may have to be switched on the right hand side of the equations if $c>>a$ to avoid losing the "1 +" due to rounding errors. Many programming languages have a specific log1p(x) function that calculates $\log_e (1+x)$ without underflow (when $x$ is small).

More generally:
:$\log _b \sum\limits_^N a_i = \log_b a_0 + \log_b \left( 1+\sum\limits_^N \frac \right) = \log _b a_0 + \log_b \left( 1+\sum\limits_^N b^ \right)$

Exponents

A useful identity involving exponents:
:$x^ = \log(x)$

or more universally:
:$x^ = a$

Other/resulting identities

:$\frac = \log_(a)$
:$\frac = \log_(a)$

Inequalities

Based on, and http://downloads.hindawi.com/archive/2013/412958.pdf

:$\frac \leq \ln(1+x) \leq \frac \leq x \mbox < x$

:\begin
\frac&\leq3-\sqrt\leq\frac \\ pt&\leq \ln(1+x)\leq \frac\leq \frac\frac \\ pt&\text 0 \le x \text < x \le 0
\end

All are accurate around $x=0$, but not for large numbers.

Calculus identities

Limits

:$\lim_\log_a(x)=-\infty\quad \mbox a > 1$

:$\lim_\log_a(x)=\infty\quad \mbox 0 < a < 1$

:$\lim_\log_a(x)=\infty\quad \mbox a > 1$

:$\lim_\log_a(x)=-\infty\quad \mbox 0 < a < 1$

:$\lim_x^b\log_a(x)=0\quad \mbox b > 0$

:$\lim_\frac=0\quad \mbox b > 0$

The last limit is often summarized as "logarithms grow more slowly than any power or root of ''x''".

Derivatives of logarithmic functions

:$\ln x = , x > 0$
:$\ln , x, = , x \neq 0$
:$\log_a x = , x > 0, a > 0, \text a\neq 1$

Integral definition

:$\ln x = \int_1^x \frac \ dt$

Integrals of logarithmic functions

:$\int \ln x \, dx = x \ln x - x + C = x(\ln x - 1) + C$
:$\int \log_a x \, dx = x \log_a x - \frac + C = \frac + C$

To remember higher integrals, it is convenient to define
:$x^ = x^(\log(x) - H_n)$
where $H_n$ is the ''n''^th harmonic number:

:$x^ = \log x$
:$x^ = x \log(x) - x$
:$x^ = x^2 \log(x) - \begin \frac \endx^2$
:$x^ = x^3 \log(x) - \begin \frac \endx^3$

Then
:$\frac\, x^ = nx^$
:$\int x^\,dx = \frac + C$

Approximating large numbers

The identities of logarithms can be used to approximate large numbers. Note that , where ''a'', ''b'', and ''c'' are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, . To get the base-10 logarithm, we would multiply 32,582,657 by , getting . We can then get .

Similarly, factorials can be approximated by summing the logarithms of the terms.

Complex logarithm identities

The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However, a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface. A single valued version, called the principal value of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single branch cut.

Definitions

In what follows, a capital first letter is used for the principal value of functions, and the lower case version is used for the multivalued function. The single valued version of definitions and identities is always given first, followed by a separate section for the multiple valued versions.

: is the standard natural logarithm of the real number ''r''.
: is the principal value of the arg function; its value is restricted to . It can be computed using .
: is the principal value of the complex logarithm function and has imaginary part in the range .
:$\operatorname(z) = \ln(, z, ) + i \operatorname(z)$
:$e^ = z$

The multiple valued version of is a set, but it is easier to write it without braces and using it in formulas follows obvious rules.

: is the set of complex numbers ''v'' which satisfy
: is the set of possible values of the arg function applied to ''z''.

When ''k'' is any integer:

:$\log(z) = \ln(, z, ) + i \arg(z)$
:$\log(z) = \operatorname(z) + 2 \pi i k$
:$e^ = z$

Constants

Principal value forms:

:$\operatorname(1) = 0$
:$\operatorname(e) = 1$

Multiple value forms, for any ''k'' an integer:

:$\log(1) = 0 + 2 \pi i k$
:$\log(e) = 1 + 2 \pi i k$

Summation

Principal value forms:

:$\operatorname(z_1) + \operatorname(z_2) = \operatorname(z_1 z_2) \pmod$
:$\operatorname(z_1) + \operatorname(z_2) = \operatorname(z_1 z_2)\quad (-\pi <\operatorname(z_1)+\operatorname(z_2)\leq \pi; \text \operatornamez_1\geq 0 \text \operatornamez_2 > 0)$
:$\operatorname(z_1) - \operatorname(z_2) = \operatorname(z_1 / z_2) \pmod$
:$\operatorname(z_1) - \operatorname(z_2) = \operatorname(z_1 / z_2) \quad (-\pi <\operatorname(z_1)-\operatorname(z_2)\leq \pi; \text \operatornamez_1\geq 0 \text \operatornamez_2 > 0)$

Multiple value forms:

:$\log(z_1) + \log(z_2) = \log(z_1 z_2)$
:$\log(z_1) - \log(z_2) = \log(z_1 / z_2)$

Powers

A complex power of a complex number can have many possible values.

Principal value form:

:$^ = e^$

:$\operatorname = z_2 \operatorname(z_1) \pmod$

Multiple value forms:

:$^ = e^$

Where , are any integers:

:$\log = z_2 \log(z_1) + 2 \pi i k_2$
:$\log = z_2 \operatorname(z_1) + z_2 2 \pi i k_1 + 2 \pi i k_2$

See also

*
*
*
*
*

References

External links

*{{sister-inline
, project=v
, links= A lesson on logarithms can be found on Wikiversity
, short=yes

Logarithm
in Mathwords

Logarithms
Mathematical identities
Articles containing proofs